Mode Theory for Optical Fiber

Transverse Electric (TE) ModesElectric field is perpendicular to the direction of propagation.Most of the magnetic field is also perpendicular to the z-direction but a small z-component exists.The ray path is meridional.It is not circular or skewed.

Transverse Magnetic (TM) ModesMagnetic field is perpendicular to the direction of propagation. Most of the electric field is also perpendicular to the z-direction but a small z-component exists.

Transverse ElectroMagnetic (TEM) ModesIn the TEM mode both the electric and magnetic fields are perpendicular to the direction of propagation i.e. z.The TEM mode is the only mode of a single-mode fibre.

Helical (Hydrid) Modes (HE and EH)In a fibre, most modes actually travel in a circular path of some kind. In this case components of both magnetic and electric fields are in the direction of propagation. Ez 0 and Hz 0.These modes are designated as either HE or EH depending on which field contributes the most to the z-direction.

Cylindrical Fiber ModesAs with dielectric waveguide TE and TM modes are common.The cylindrical fiber is bound in two directions hence two integers l and m are necessary to specify the modes.For cylindrical waveguides we refer to TElm, TMlm, HElm and EHlm modes.The EH and HE modes are due to nonzero Ez and Hz components.

OF support weakly guiding approximation, where Relative RI difference () is very small as compared to unity.

For these guiding structures the mode theory gives dominant transverse field components, i.e. the approximate solutions for the full set of HE, EH, TE and TM modes may be given by two linearly polarized components called LP modes.LP modes also use subscripts l and m. 2l field maxima exist around circumference of core and m field maxima along radius vector.

Correspondence between LP modes and traditional exact modesLinearly Polarized ModeExact ModesLP01 HE11 LP11 HE21,TE01, TM01 LP21 HE31, EH11 LP02HE12LP31HE41, EH21LP12HE22, TE02, TM02LPlmHE2m, TE0m, TM0mLPlm(l>1)HEl+1,m, EHl-1,m

Energy Distribution of Some LP Modes in Fiber

Using the previous equation in It is the Laplacian operater. In cylindrical coordinate system it is given as:

The propagation constant of the guided mode, lies in the rangen2k < < n1kSolution of the wave equation for cylindrical fiber are is given in the form

Using weakly guiding approximation we can write the scalar wave equation for the cylindrical homogeneous core as:

Above represents the dominant electric field component.The periodic dependence on following cos and sin functions gives a mode of radial order l.The fiber supports a finite number of guided modes of the form given by above equation.

If we introduce the solution given by this equation in wave equation we get a differential equation:

The above equation is a Bessel differential equation, for core region the solutions are Bessel function denoted by Jl. A graph of this with r is shown in the next slide

The field is finite at r = 0 and it is represented by zero order Bessel function J0As r increases and goes towards infinity the field vanishes and in the cladding the solution is given by another function called Modified Bessel functions denoted by Kl. U is Radial propagation parameterW is Cladding decay parameterR=r/a

Graph of Modified bessel function Kl(r) against r for l=0 & 1

U and W are equal to the eigenvalues in the core and cladding respectively defined as U = a(n12k2 2)1/2W = a(2 n22k2)1/2

A very useful relation using U and W can be given byV = (U2 + W2)1/2V = ka(n12 n22)1/2V is called the normalized frequency or V number.It is a dimensionless parameter.Expression relates core radius, relative reflective index difference () and wavelength.

Another important parameter can also define is the normalized propagation constant b for a fiber as

b = 1 (U2/V2)b = [(/k)2 n22] / (n12 n22) b = [(/k)2 n22] / 2n12

The limit of is between n1k and n2k but b lies between 0 - 1

When you have U and as function of V, propagation characteristics of modes and their dependence on frequency and fiber geometry can be obtained. When = n2k, then mode is no longer properly guided.In this case mode is said to be cutoff and W = 0

Unguided or radiating modes have frequencies below cutoff where < n2k and W is imaginary. These modes are also called leaky modes.As moves toward n2k less and less power is propagated in cladding and when as = n1k all power is confined to core.Range of signifies the value for guided modes in the fiber

Vc is different for different modes.First zero crossing of J1 occurs at V = 0 and this corresponds to cutoff for LP01 mode.The first zero crossing for J0 is when V = 2.405 giving Vc for LP11 mode.Second zero crossing for J1 occurs at V = 3.83 giving Vc for LP02 of 3.83

Hence we can have fibers manufactured with a particular V no. for limiting certain modes.

It may please be further noted that cutoff value of V i.e. Vc occurs when = n2k correspond to b = 0